We consider a system of two kinetic equations coupled by non-local interaction terms. This system was recently obtained as the mean-field limit of a second-order system of two non-locally interacting species. Models of this type are encountered in a variety of contexts and are typically used to describe large systems of indistinguishable agents such as cell colonies, flocks of birds, schools of fish, flocks of sheep, etc.
To simulate this kinetic model, we propose an upwind finite volume method. This method is constructed in such a way that mass is preserved and positivity is maintained. Moreover, convex functionals of the discrete solution are controlled, which we use to show the convergence of the method. Under additional regularity assumptions, we provide explicit error estimates. Finally, we present simulations that show the behaviour of the system numerically.
Joint work with Valeria Iorio and Markus Schmidtchen.

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